Moments of Random Vectors Which Belong to Some Domain of Normal Attraction

Abstract

Let $X$ be a random vector on $\mathbb{R}^k$ whose distribution $\mu$ belongs to the domain of normal attraction of some operator stable law $\nu$. For a given $\nu$ it has been shown elsewhere that for certain ranges of $\alpha$ depending on $\nu$, either $E|\langle X, \theta\rangle|^\alpha$ is finite for every $\theta \neq 0$ or is infinite for every $\theta \neq 0$. In this paper we show that the set of $\alpha$ for which $E|\langle X, \theta\rangle|^\alpha$ exists depends, in general, on both $\theta$ and $\nu$, and we obtain a complete description of the cases in which $E|\langle X, \theta\rangle|^\alpha$ can be guaranteed either to exist or to diverge, just on the basis of $\theta$ and $\nu$.